Non-algebraic Galois Extension

Say is Galois*.
Must it be the case that is algebraic?
(This was not addressed in my book).

*)I just realized how bad my question is. A Galois extension is an algebraic extension such that . Note we use "algebraic" within the definition itself. My question would be more appropriately asked if then must it follow that is algebraic?

this is a good question! the answer is No! because, for example, if is transcendental over and , then

it can be proved (not very easily though!) that the main part of the proof is to show that

consists of all maps defined by where and

I actually was thinking about , but I never did any of the details to prove it is Galois.

Now this leads me to my next question: is there anything interesting about a non-algebraic Galois extension?
(Maybe my books talk about this stuff near the end on transcendental extensions,
but it would be interesting to me to know ahead of time).